Easy Ways to Find the Area Under a Curve

Finding the area under a curve is an important part of calculus. There are different methods to do this:

1. Riemann Sums:

   • You use rectangles to estimate the area.
   • There are three types: left sums, right sums, and midpoint sums.
   • If you use more rectangles (let’s call that $n$), your estimate gets better.

2. Trapezoidal Rule:

   • This method uses trapezoids instead of rectangles to find the area.
   • Here’s a simple formula to remember:
   $$A \approx \frac{b-a}{2n} \left(f(a) + 2\sum_{i=1}^{n-1}f(x_i) + f(b)\right)$$
   • It usually gives a more accurate answer than Riemann sums when $n$ is large.

3. Definite Integrals:

   • This method tells you the exact area under the curve from point $a$ to point $b$.
   • You can write it like this:
   $$A = \int_a^b f(x) \, dx$$
   • This gives you the exact area, using an important concept in calculus called the Fundamental Theorem of Calculus.

These methods help us find both approximate and precise areas under curves in many different situations.